Review Flashcards
E(X)
Nb. E(X) = μx
E{g(X)}
e.g. g(X) = x2
The three expected value rules
- E(X+Y+Z) = E(X) + E(Y) + E(Z)
- E(bX) = bE(X)
- E(b) = b
Two formulae for population vairance of a discrete random variable, σ2 using
- var(X) = σ2 = E {(X - μx)2}
- var(X) = σ2 = E (X2) - μx2
cov (X,Y)
If X and Y are independent
then E[g(X)h(Y)] =
E[g(X)h(Y)] = E[g(X)] E[h(Y)]
Nb. This means that if X and Y are independent then E(XY) = E(X) E(Y)
If X is a random variable with X = μx + u where u is a pure random component, what is the expectation and variance of u?
E(u)=0 and σu2 = σX2 = E(u2)
Covariance of X and Y
Cov(X,Y)= σXY = E{(X-μX)(Y-μY)}
1) If Y = V+W, Cov (X,Y) =
2) If Y=bZ, Cov (X,Y) =
3) If Y = V + b, Cov (X,Y)=
1) If Y = V+W, Cov (X,Y) = Cov(X,V) + Cov (X,W)
2) If Y=bZ, Cov (X,Y) = bCov(X,Z)
3) If Y = V + b, Cov (X,Y) = 0
Formula for the sample covariance of X and Y
sXY = Σ(xi-x̄)(yi-ȳ)/(n-1)
If Y = V + W, var(Y) =
If Y=bZ, then var(Y) =
If Y=b, then var(Y) =
If Y = V + W, var(Y) = var(V) + var(W) +2cov(X,Y)
If Y=bZ, then var(Y) = b2var(Z)
If Y=b, then var(Y) = 0
Prove the covariance of X and Y is zero if X and Y are independent variables
If X and Y are independent,
Cov(X,Y) = E{(X-μx)(Y-μY)} = E(X-μx)E(Y-μY)
= (E(X)-μx)(E(Y)-μY)
= (μx-μx)(μY-μY) = 0
Population correlation coefficient
ρXY = σXY / √(σX2 + σ2Y)
E(x̄) =
σx̄2 =
var(x̄)=σX2/n
Bias is
Bias is the difference between an estimator’s expected value and the value of the population characteristic
One estimator is more efficient than another if
One estimator is more efficient than another if it is more likely to give an accurate estimate, i.e. its probability density function is more concentrated around the true value - it has the smaller variance out of two unbiased estimators
Mean square error of an estimator Z of parameter θ
- two definitions
MSE (Z) = E{(Z-θ)2}
MSE (Z) = Var(Z) + bias2
